Symplectic Bregman divergences (2408.12961v3)

Abstract: We present a generalization of Bregman divergences in symplectic vector spaces that we term symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symplectic generalization of the Fenchel-Young inequality which relies on the notion of symplectic subdifferentials. The symplectic Fenchel-Young inequality is obtained using the symplectic Fenchel transform which is defined with respect to the symplectic form. Since symplectic forms can be generically built from pairings of dual systems, we get a generalization of Bregman divergences in dual systems obtained by equivalent symplectic Bregman divergences. In particular, when the symplectic form is derived from an inner product, we show that the corresponding symplectic Bregman divergences amount to ordinary Bregman divergences with respect to composite inner products. Some potential applications of symplectic divergences in geometric mechanics, information geometry, and learning dynamics in machine learning are touched upon.

• The paper introduces symplectic Bregman divergences by extending traditional divergence measures using a generalized Fenchel-Young inequality in symplectic spaces.
• It demonstrates that classical Bregman divergences can be recovered when the symplectic form is induced by an inner product, bridging the gap between dual systems.
• It outlines potential applications in geometric mechanics, information geometry, and machine learning, offering new opportunities for optimization and modeling.

Symplectic Bregman Divergences: An Expert Overview

The paper "Symplectic Bregman Divergences" by Frank Nielsen presents an advanced generalization of Bregman divergences within symplectic vector spaces and explores the theoretical underpinnings and potential applications of this new class of divergences. The core contribution is the introduction of symplectic Bregman divergences, building on the symplectic generalization of the Fenchel-Young inequality and leveraging the notion of symplectic subdifferentials.

Key Concepts and Definitions

Symplectic geometry forms the backbone of this paper. Historically rooted in the works of Lagrange and further developed in classical and quantum mechanics, symplectic geometry describes a structure on vector spaces that is fundamentally non-Euclidean and involves a symplectic form, a bilinear map that is skew-symmetric and non-degenerate. This form is central to the definition of symplectic Bregman divergences.

Symplectic Fenchel Transform and Inequality

A pivotal element in defining symplectic Bregman divergences is the symplectic Fenchel transform. This transform generalizes the classical Fenchel transformation by optimizing with respect to a symplectic form. The corresponding symplectic Fenchel-Young inequality provides the foundation for defining symplectic Fenchel-Young divergences, which measure dissimilarity in a manner akin to conventional Fenchel-Young divergences but within the symplectic framework.

Core Contributions

Symplectic Bregman Divergences

The primary contribution is the definition of symplectic Bregman divergences. These divergences are formulated based on a smooth convex potential function $F$ and relate changes in this function to a symplectic form. Mathematically, the symplectic Bregman divergence $B_F^\omega(z_1: z_2)$ between two points $z_1$ and $z_2$ in a symplectic vector space $(Z, \omega)$ is given by:

$$B_F^\omega(z_1:z_2) = F(z_1) - F(z_2) - \omega(\nabla^\omega F(z_2), z_1 - z_2)$$

Special Cases and Dual Systems

The paper demonstrates that when the symplectic form is induced by an inner product on $X$, the symplectic Bregman divergence recovers the traditional Bregman divergence with respect to a composite inner product. This extension allows symplectic Bregman divergences to define Bregman divergences on dual systems equipped with pairing products.

Implications and Future Directions

Practical and Theoretical Implications

Understanding and utilizing symplectic Bregman divergences can have significant implications in various fields, including geometric mechanics, information geometry, and machine learning. For example:

• Geometric Mechanics: They can model systems with dissipative terms, offering a new perspective on mechanics that can bridge classical and quantum paradigms.
• Information Geometry: These divergences can enrich the geometry of statistical manifolds, providing new tools for understanding complex systems.
• Machine Learning: Symplectic geometry has already shown promise in accelerating optimization methods and enhancing physics-informed neural networks (PINNs).

Speculative Future Developments

Future research can explore several avenues building on this foundation:

1. Algorithmic Development: Algorithms harnessing symplectic Bregman divergences could be developed for efficient optimization in high-dimensional spaces, potentially improving training dynamics in deep learning.
2. Extended Applications: Further exploration of symplectic structures in machine learning could lead to innovative approaches in reinforcement learning, variational inference, and beyond.
3. Interdisciplinary Research: Integrating insights from geometric mechanics with machine learning could lead to new theories and applications in computational physics and AI-driven scientific discovery.

Conclusion

The introduction of symplectic Bregman divergences represents a meaningful theoretical expansion in the paper of divergences. By incorporating symplectic forms and extending classical concepts such as the Fenchel-Young inequality into the symplectic domain, this paper opens the door for new methodologies and applications across multiple disciplines. The potential of these divergences to influence the trajectory of geometric mechanics, information geometry, and machine learning underlines the promising horizon of this research.